A $C$-semigroup (respectively a $D$-semigroup) is positioned numerical semigroup $S$ such that $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-1}{2}$ $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-2}{2}$). In this paper we study these semigroups giving formulas for the Frobenius number, pseudo-Frobenius and type. Furthermore, give algorithms computing whole set of $C$-semigroups $D$-semigroups.

1. Background and introduction. In his 1984 Memoir of the American Mathematical Society, Andrews [2] introduced two families of partition functions, φk(m) and cφk(m), which he called generalized Frobenius partition functions. In this paper, we will focus our attention on one of these functions, namely cφ2(m), which denotes the number of generalized Frobenius partitions of m with 2 colors. In [2...

Frobenius bimodules are connected with Frobenius algebras and extensions. For instance, a ring extension φ : R → S is a Frobenius extension if and only if RSS is a Frobenius bimodule [1]. Brzeziński and Gómez-Torrecillas studied in [4] certain properties of comatrix corings in relation to properties of bimodules. In particular they showed that the comatrix coring [6] induced by any Frobenius bi...

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optim...

‎the frobenius complement of a given frobenius group acts on its kernel‎. ‎the scheme which is arisen from the orbitals of this action is called ferrero pair scheme‎. ‎in this paper‎, ‎we show that the fibers of a ferrero pair scheme consist of exactly one singleton fiber and every two fibers with more than one point have the same cardinality‎. ‎moreover‎, ‎it is shown that the restriction of a...

We consider the Diophantine problem of Frobenius for semigroup S ( d ) where d denotes the tuple (d1, d2, d3), gcd(d1, d2, d3) = 1. Based on the Hadamard product of analytic functions [17] we have found the analytic representation for the diagonal elements akk ( d ) of the Johnson’s matrix of minimal relations [12] in terms of d1, d2, d3. Bearing in mind the results of the recent paper [10] thi...

This paper consists of three results on Frobenius categories: (1)we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen–Macaulaymodules over some additive category; this is an analogue of Gabriel–Quillen’...

Recall that a finite-dimensional commutative associative algebra equipped with an invariant nondegenerate symmetric bilinear form is called a Frobenius algebra (here, we do not require an existence of a unit in Frobenius algebra). Any commutative quasi-Frobenius algebra is always Frobenius, i.e., if the identity ab = ba (commutativity) is fulfilled in a quasi-Frobenius algebra, then the identities